is $\delta$-compact set complete?

Question

We define $\delta$-compact metric space as monotone union of compact sets. $M=\bigcup M_i$ ($M_i\subset M_{i+1}$), is it complete?

Answer 1

Try $(0,1) = \bigcup_n [1/n, 1 - 1/n]$.

Answer 2

Completeness is a metric property. This may not even be true for $\mathbb R$ with the standard topology if you don't use the standard metric.